\section{Discontinuous Galerkin formulation of Poisson Problem}

\begin{equation}\nonumber
\begin{split}
\nabla^2 u = f \Rightarrow 
\begin{cases}  
\nabla u = \textbf{q}\\
\nabla \cdot \textbf{q} = f
\end{cases} &\Rightarrow
\begin{cases}
\displaystyle{\int_{\kappa}} <\nabla u,\textbf{v}> = \displaystyle{\int_{\kappa}} <\textbf{q}, \textbf{v}>\\
\displaystyle{\int_{\kappa}} w\nabla\cdot\textbf{q} = \displaystyle{\int_{\kappa}} fw
\end{cases}\\
\Rightarrow
\begin{cases}
-\displaystyle{\int_{\kappa}}u\nabla\cdot\textbf{v} + \oint_{\partial\kappa}u<\textbf{v},\textbf{n}> = \displaystyle{\int_{\kappa}}<\textbf{q},\textbf{v}>\\
-\displaystyle{\int_{\kappa}}<\nabla w, \textbf{q}> + \oint_{\partial\kappa}w<\textbf{q},\textbf{n}> = \displaystyle{\int_{\kappa}}fw
\end{cases} &\Rightarrow
\begin{cases}
-\displaystyle{\int_{\Omega}}u\nabla\cdot\textbf{v} + \displaystyle{\sum_{\aleph\{\kappa\}}}\oint_{\partial\kappa}u<\textbf{v},\textbf{n}> = \int_{\Omega}<\textbf{q},\textbf{v}>\\
-\displaystyle{\int_{\Omega}}<\nabla w, \textbf{q}> + \displaystyle{\sum_{\aleph\{\kappa\}}}\oint_{\partial\kappa}w<\textbf{q},\textbf{n}> = \int_{\Omega}fw
\end{cases}
\end{split}
\end{equation}

By introducing operator $\{\!\!\{\cdot\}\!\!\}$ and $[\![\cdot]\!]$

\begin{equation}\nonumber
\begin{split}
\{\!\!\{u\}\!\!\} &= \frac{1}{2}(u^{+}+u^{-})             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;          \{\!\!\{\textbf{q}\}\!\!\} = \frac{1}{2}(\textbf{q}^{+}+\textbf{q}^{-})\\
[\![u]\!] &= u^{+}\textbf{n}^{+} + u^{-}\textbf{n}^{-}             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;          [\![\textbf{q}]\!] = \textbf{q}^{+}\cdot\textbf{n}^{+} + \textbf{q}^{-}\cdot\textbf{n}^{-}
\end{split}
\end{equation}

we have

\begin{equation}\nonumber
\begin{split}
\begin{cases}
\displaystyle{\sum_{\aleph\{\kappa\}}}\oint_{\partial\kappa}u<\textbf{v},\textbf{n}> = \int_{\Gamma_{\mathcal{I}}}\{\!\!\{u\}\!\!\}[\![\textbf{v}]\!] + \{\!\!\{\textbf{v}\}\!\!\}[\![u]\!]\\
\displaystyle{\sum_{\aleph\{\kappa\}}}\oint_{\partial\kappa}w<\textbf{q},\textbf{n}> = \int_{\Gamma_{\mathcal{I}}}\{\!\!\{w\}\!\!\}[\![\textbf{q}]\!] + \{\!\!\{\textbf{q}\}\!\!\}[\![w]\!]
\end{cases}\\
\Rightarrow
\begin{cases}
-\displaystyle{\int_{\Omega}}u\nabla\cdot\textbf{v} + \int_{\Gamma_{\mathcal{I}}}(\{\!\!\{u\}\!\!\}[\![\textbf{v}]\!] + \{\!\!\{\textbf{v}\}\!\!\}[\![u]\!]) = \int_{\Omega}<\textbf{q},\textbf{v}>\\
-\displaystyle{\int_{\Omega}}<\nabla w, \textbf{q}> + \int_{\Gamma_{\mathcal{I}}}(\{\!\!\{w\}\!\!\}[\![\textbf{q}]\!] + \{\!\!\{\textbf{q}\}\!\!\}[\![w]\!]) = \int_{\Omega}fw
\end{cases}
\end{split}
\end{equation}

Approximate the unknown using piecewise continuous basis functions

\begin{equation}\nonumber
\begin{split}
&u = \sum_{m}\alpha_{m}\phi_{m}, \textbf{q} = \sum_{n}\beta_{n}\boldsymbol{\psi}_{n} \Rightarrow 
\begin{cases}
-\displaystyle{\int_{\Omega}}\sum_{i}\alpha_{i}\phi_{i}\nabla\cdot\boldsymbol{\psi}_{j} + \mathbb{X} = \int_{\Omega}<\sum_{s}\beta_{s}\boldsymbol{\psi}_{s},\boldsymbol{\psi}_{j}>\\
-\displaystyle{\int_{\Omega}}<\nabla \phi_{j}, \sum_{i}\beta_{i}\boldsymbol{\psi}_{i}> + \mathbb{Y} = \int_{\Omega}f\phi_{j}
\end{cases}\\
&\Rightarrow
\begin{cases}
-\sum_{i}\alpha_{i}\displaystyle{\int_{\Omega}}\phi_{i}\nabla\cdot\boldsymbol{\psi}_{j} + \mathbb{X} = \sum_{s}\beta_{s}\int_{\Omega}<\boldsymbol{\psi}_{s},\boldsymbol{\psi}_{j}>\\
-\sum_{i}\beta_{i}\displaystyle{\int_{\Omega}}<\nabla \phi_{j}, \boldsymbol{\psi}_{i}> + \mathbb{Y} = \int_{\Omega}f\phi_{j}
\end{cases}
\Rightarrow
\begin{cases}
-\textbf{A}_{1,1}\boldsymbol{\alpha} + \mathbb{X} = \textbf{A}_{1,2}\boldsymbol{\beta}\\
-\textbf{A}_{2,2}\boldsymbol{\beta} + \mathbb{Y} = \textbf{b}
\end{cases}\\
&\Rightarrow \begin{pmatrix} \textbf{A}_{1,1} & \textbf{A}_{1,2} \\ \textbf{0} & -\textbf{A}_{2,2} \end{pmatrix} \begin{pmatrix} \boldsymbol{\alpha} \\ \boldsymbol{\beta} \end{pmatrix} + \begin{pmatrix} -\mathbb{X} \\ \mathbb{Y} \end{pmatrix} = \begin{pmatrix} \textbf{0} \\ \textbf{b} \end{pmatrix}
\end{split}
\end{equation}